Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium Julian Fischer Max Planck Institute for Mathematics in the Sciences, Leipzig Title: A higher-order large-scale regularity theory for random elliptic operators Abstract: We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. Under the assumptions of stationarity and slightly quantified ergodicity of the ensemble, we derive a $C^{k,\alpha}$-"excess decay" estimate on large scales and a $C^{k,\alpha}$-Liouville principle for any $k\geq 2$: For a given $a$-harmonic function $u$ on a ball $B_R$, we show that its energy distance on some ball $B_r$ to the space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has the natural decay in the radius $r$, at least above some minimal (random) radius $r_0$. The Liouville principle states that the space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has (almost surely) the same dimension as in the constant-coefficient case. Our results rely on the existence of higher-order correctors for the homogenization problem, which we establish by an iterative construction.Joint work with Felix Otto.Date: Tuesday, May 3, 2016Time: 1:30 pmLocation: Wean Hall 7218Submitted by:  David Kinderlehrer